Modelling of multi‐block data

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Abstract
Here is presented a unified approach to modelling multi‐block regression data. The starting point is a partition of the data X into L data blocks, X = (X1, X2,…,XL), and the data Y into M data‐blocks, Y = (Y1, Y2,…,YM). The methods of linear regression, X→Y, are extended to the case of a linear relationship between each Xi and Yj, Xi→Yj. A modelling strategy is used to decide if the residual Xi should take part in the modelling of one or more Yjs. At each step the procedure of finding score vectors is based on well‐defined optimisation procedures. The principle of optimisation is based on that the score vectors should give the sizes of the resulting Yjs loading vectors as large as possible. The partition of X and Y are independent of each other. The choice of Yj can be Xj, Yi = Xi, thus including the possibility of modelling X→Xi, i = 1,…,L. It is shown how these methods can be extended to a network of data blocks. Examples of the optimisation procedures in a network are shown. The examples chosen are the ones that are useful to work within industrial production environments. The methods are illustrated by simulated data and data from cement production. Copyright © 2007 John Wiley & Sons, Ltd.

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